Watkins argues that this phrase represents an Indo-European textual formula. However, Margalit Finkelberg, in "More on κλέος ἄφθιτον", Classical Quarterly 57.2 (2007), argues that śrávas and kléos were not always synonymous, and that the "fame" sense developed independently in both languages.
*ḱlewos is built on *ḱleu- "to hear" - that is, "what is heard about someone, fame". *n̥dʰgʷʰitom "imperishable" consists of a negative prefix *n̥- and *gʷhðei(ǝ)- "to perish, destroy" (nowadays written as *dʰgʷʰei-, I assume). It has no English descendents, but, in addition to Greek and Indo-Iranian, is probably found in Latin situs "mud, dirt".
The Vedic phrase is from R̥gveda 1.9.7 in the form śrávo... ákṣitam.
सं गोमदिन्द्र वाजवदस्मे पृथु श्रवो बृहत् | विश्वायुधेह्यक्षितम् ||
maṃ gomadindra vājavadasme pṛthu śravo bṛhat
Give, Indra, wide and lofty fame, wealthy in cattle and in strength, Lasting our life-time, failing not.
śravas is a neuter noun. I assume it takes the form śravo here because of sandhi. In The American Heritage Dictionary of Indo-European Roots Watkins cites it as śravaḥ.
The phrase in the Iliad, from Perseus:
Iliad book 9, lines 412-415
εἰ μέν κ' αὖθι μένων Τρώων πόλιν ἀμφιμάχωμαι
ὤλετο μέν μοι νόστος, ἀτὰρ κλέος ἄφθιτον ἔσται:
εἰ δέ κεν οἴκαδ' ἵκωμι φίλην ἐς πατρίδα γαῖαν,
ὤλετό μοι κλέος ἐσθλόν, ἐπὶ δηρὸν δέ μοι αἰὼν
ei men k' authi menōn Trōōn polin amphimakhōmai
ōleto men moi nostos, atar kleos aphthiton estai:
ei de ken oikad' hikōmi philēn es patrida gaian,
ōleto moi kleos esthlon, epi dēron de moi aiōn
if I abide here and war about the city of the Trojans,
then lost is my home-return, but my renown shall be imperishable;
but if I return home to my dear native land,
lost then is my glorious renown, yet shall my life long endure